MATH 223 – FINAL EXAM
APRIL, 2005
Instructions:
(a) There are 10 problems in this exam. Each problem is worth five points,
divided equally among parts.
(b) Full credit is given to complete work only. Simply writing down an answer is
not enough (unless told otherwise).
(c) No calculators, books or notes are allowed.
(d) All vector spaces are assumed to be real vector spaces. No complex numbers
will be needed.
(e) The usual notation is assumed:
• Mm×n is the space of m × n matrices.
• Pn is the space of polynomials in one variable of degree n or less.
• P is the space of all polynomials in one variable.
(f) Good luck!
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MATH 223 – FINAL EXAM
Problem 1. Which of the following sets W are subspaces? No proof is necessary,
although you may want to prove it to convince yourself.
(a) W ⊂ P4 the set of palindromic polynomials:
W = {a + bx + cx2 + bx3 + ax4 |a, b, c ∈ R}.
(b) W ⊂ M2×2 the set of non-invertible matrices:
W = {A ∈ M2×2 | det(A) = 0}.
(c) Given linear transformations T : U → V and S : U → V , let W ⊂ V be the set
of all vectors ~v ∈ V that can be expressed as
~v = T (~u1 ) + S(~u2 )
for some vectors ~u1 , ~u2 ∈ U .
(d) W ⊂ R3 is the intersection of two cylinders defined by the equations (x − 1)2 +
y 2 = 1 and, (x + 1)2 + y 2 = 1:
W = {(x, y, z)| (x − 1)2 + y 2 = 1,
(x + 1)2 + y 2 = 1}.
(Hint: draw a picture of the cylinders.)
(e) W ⊂ Mn×n the set of all matrices having the vector
 
1
 
2

~v = 
 .. 
.
 
n
as an eigenvector (with arbitrary eigenvalue).
MATH 223 – FINAL EXAM
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Problem 2. Which of the following functions T are linear transformations? Again,
no proof is necessary.
(a) T : P3 → P6 is given by T (f (x)) = f (x2 ).
(b) T : M3×3 → M3×3 that adds the first column to the last one:
T [~a1 |~a2 |~a3 ] = [~a1 |~a2 |~a1 + ~a3 ].
(c) T : P → R that maps the zero polynomial to zero and a nonzero polynomial to
its last nonzero coefficient:
T (a0 + a1 x + . . . + an xn ) = an ,
where an 6= 0.
(d) T : Mn×n → Pn that maps a matrix A to its characteristic polynomial fA (t).
(e) T : R → R given by T (x) = 5x + 3.
Problem 3. Note that if ~v , w
~ ∈ Rn are column vectors, then the product of matrices
~v · w
~ t is an n × n matrix. Let
 
1
 
2

w
~ =
 ..  .
.
 
n
If ~v1 , . . . , ~vn is a basis of Rn , show that the set of matrices
S = {~v1 · w
~ t , ~v2 · w
~ t , . . . , ~vn · w
~ t}
is linearly independent. (Hint: What are the columns of the matrices in S?)
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MATH 223 – FINAL EXAM
Problem 4. Let T : R5 → R4 be the linear transformation
 


x1
 
x
−
2x
+
2x
+
4x
1
2
3
5
x  

 2 
   −x1 + 2x2 + 2x3 + x4 + 7x5 


T
.
x3  = 

  
x
+
3x
3
4

x4  
 
2x1 − 4x2 − x3 + 3x4 − 10x5
x5
(a) Find a basis for the null-space of T .
(b) Find a basis for the range of T .
Problem 5. Let V be the vector space of all continuous functions f : R → R and
let W ⊂ V be the subspace spanned by
ex , xex , x2 ex .
You may assume without proof that these three functions are linearly independent.
Let T : W → W be the derivative d/dx. Find the inverse of T . (Note: you can find
the inverse by inverting the matrix of T in some basis, but your final answer should
be in the form
T −1 (aex + bxex + cx2 ex ) = (a + b + 2c)ex + . . . + (. . .)x2 ex .
Since T is the derivative, its inverse can be found by integration. However, only
minimal credit will be given for such a calculus proof.)
Problem 6. Let A be a nonzero 10 × 10 matrix such that A25 = 0.
(a) Show that 0 is an eigenvalue of A; in other words, there is a corresponding
eigenvector ~v .
(b) Show that A has no other eigenvalues.
(c) Show that A is not diagonalizable.
MATH 223 – FINAL EXAM
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Problem 7. A co-op produces four types of cookies: A, B, C, and D. Ingredients
needed to make one box of cookies of each type are given in the table below:
A B C D
flour (cups)
3
6
3
9
butter (lb.)
1
2
1
3
sugar (cups)
2
1
3
8
eggs
1
4
2
7
chocolate (lb.) 0
3
1
2
During one hour of operation, the following amount of ingredients were used: 33 cups
of flour, 11 lb. of butter, 23 cups of sugar, 21 eggs, 7 lb. of chocolate. Find how
many boxes of each type were produced.
Problem 8. Let An be the n × n matrix below with non-zero entries on the three
diagonals only:


6 1


5 6 1






 5 6 1


An = 
.


...






5
6
1


5 6
(a) Use expansion along the first row and first column to express det(An ) in terms
of det(An−1 ) and det(An−2 ):
det(An ) = a det(An−1 ) + b det(An−2 ).
Use this formula to find det(A4 ). (det(A4 ) is a big number, close to 1000.)
(b) Use the formula from the previous part to give an exact expression for det(An ).
(If you could not determine a and b in part (a), take a = 4, b = 5.)
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MATH 223 – FINAL EXAM
Problem 9. Let A be a p × m matrix and B a m × n matrix.
(a) If Rank(A) = m and Rank(B) = n, find Rank(AB). (Give a complete argument. A number is not enough.)
(b) If Rank(A) = Rank(B) = m, find Rank(AB). (Again, a complete proof is
required.)
Problem 10. Consider a population model with three populations Xn , Yn , and Zn
at year n. The change in the populations is described by the model
Xn+1 = 2Xn +1.5Yn −3Zn
Yn+1 =
6.5Yn −9Zn
Zn+1 =
3Yn −4Zn
Describe the behavior of this population model as n → ∞: find all initial conditions
X0 , Y0 , Z0 such that the populations grow to infinity, and all initial conditions such
that the populations die out. (Do not worry about populations being negative.)
End of exam.